New definitions in the OpenAI math papers

14 October 2026

Attribution: Experiments were performed by Claude Opus 5.5, and the writeup below is largely produced by the same model. David checked a very small sample of the papers by hand.

Substantial definitions appear far less often in the OpenAI release than in average papers by human mathematicians. After a full reading by an LLM, 2 of the 719 AI-generated manuscripts in OpenAI's 6 October 2026 release introduce a notable new definition (0.3%), against 6 of 113 random recent arXiv math papers (5.3%) judged by the same blind protocol. One further AI candidate, the "tensor condition" for Euclidean Ramsey sets, could be significant if its unverified proof holds. The AI papers also acknowledge the closest existing version of their new notions less reliably.

AI versus human mathematicians

Under one identical blind protocol, random arXiv math papers introduce a notable new definition about 15 times as often as the OpenAI papers (Fisher exact p = 0.0001).

Every paper was screened blind by GPT-6 Sol, and every paper Sol flagged was then read in full by a Claude Opus agent with a literature search; details are under Method.

• The two AI notables are both at the low end: "compatible coordinates" (Koebe's circle domain conjecture) and an "independently translated elliptic capacity" (Fourier restriction). Adding the two AI candidates this protocol missed, the tensor condition and readable label trees, gives 4 of 719 (0.6%), still far below arXiv (p = 0.0007).
• The six arXiv notables are spread across operator algebras (two), algebraic geometry, complex dynamics, number theory and integer optimisation. We do not name the human papers, here or elsewhere, since the verdicts are one automated reading rather than a judgement anyone should be held to.
• The arXiv sample is all of mathematics, including applied and numerical papers that tend to make new definitions less often, whereas every OpenAI paper attacks a pure-mathematics problem; so the gap probably understates the difference in theory-building.

Citing prior work

The AI papers fully acknowledge the closest existing version of their new notion in 6 of 19 checked cases, against 8 of 10 for arXiv (Fisher exact p = 0.02). For each paper it assessed, the Opus agent recorded whether the paper cites the closest prior notion it found and explains the relationship.

• The two outright misses are both AI. "Property L" in the proof that every K3 surface is Oka is essentially Alarcón–Forstnerič's local spray approximation property (2023), which the paper does not cite. The finiteness framework in the planar limit-cycle paper repackages the route of Wilkie's theorem of the complement, also uncited.
• The typical AI "partly" cites the right papers but never says the new notion is a special case or variant of something in them, such as tree norms that are the base norm of a 1990 tree cone, or a capacity that is a variant of the Wang–Zahl convex Wolff axioms.
• Humans miss too. One arXiv paper re-defines a notion that had appeared shortly before, citing that work only generically.
• The check is strict: "partly" often means a secondary precursor is missing. Samples are small, so treat this as a pattern worth a closer look rather than a measured rate.

The strongest AI candidates

Tensor condition (family 172). A finite set is Ramsey if and only if it is "spherical before multiplication". There must be a matrix over F ⊗_Q F whose quadratic evaluations at every point vanish in F ⊗_Q F itself, not merely after multiplying out to F, while its multiplied spatial block is the identity. Opus calls it natural rather than ad hoc: in effect the universal completion of the classical invariant-and-colouring method. If the long sufficiency proof is right, it settles a decades-old classification in exactly the form a new standard notion takes. Sol rated it "none" because it is a criterion, not a new object; that disagreement is itself a data point about what "new definition" should mean. A precursor, Pálvölgyi's A cyclic non-Ramsey heptagon, was also written entirely by ChatGPT; it uses related certificates but states no such criterion.

Readable label tree (family 107). A partition of a tensor's support down to single monomials, such that at each node the next branch can be read off from either of two chosen sides. Each side's accumulated conditional entropy becomes a matrix dimension. It is a new variant of laser-method partitions, but thin: its force comes from a separation lemma. The headline claim ω < 2.258 (the record is 2.371) is extraordinary and unverified.

What the other AI definitions look like

The other AI definitions Opus examined are respectable but derivative, and fall into three kinds:

• Next member of an existing family: the critical LQG sphere, following the critical LQG disk of Aru–Holden–Powell–Sun; massive SLE at κ = 16/3, a case Makarov–Smirnov named without a drift formula.
• An existing theory with a new generating class: Woolf's Witt groups of constructible sheaves, regenerated from arbitrary compact sources so the Hilbert–Smith argument can pass through wild orbit spaces.
• Axioms engineered for one theorem: the several-variable quasianalytic class, the "visible" spectral subspace, the limit-cycle "packets". Each assumes exactly the stability its theorem needs.

Several sit close to prior work the papers do not cite, such as Wilkie's theorem of the complement for the limit-cycle paper. The assessors kept reaching the same verdict: a human paper with these results would be remembered for the theorem, not the definition.

Caveats

• Partial reading in the screens. Luna and Sol saw only each paper's opening and extracted definitions, so a concept introduced deep in a long paper could slip through. The 0-of-40 control result makes this unlikely to be common, not impossible.
• Novelty rests on the models' knowledge of the literature. Sol and Opus repeatedly found close precursors, so if anything the verdicts overrate novelty.
• No proofs were checked. Several concepts are only as valuable as unrefereed proofs make them.
• The corpus selects for solving. OpenAI posed about 4,000 existing problems and kept the results. That setup rewards answering stated questions, not building theory, so the finding describes this release as much as what AI can do.

If you have read any of the OpenAI papers and think we missed or misjudged a genuinely new definition, I would be glad to hear about it.

Method

The comparison above comes from a blind pass over both corpora. An earlier first pass screened the release alone and picked out the AI candidates discussed above.

Blind comparison of both corpora

Protocol, applied to both corpora: each paper's TeX was cut to a digest with author names, citation keys, acknowledgements and addresses stripped; GPT-6 Sol (high effort) judged every digest with a neutral prompt that does not mention AI; every paper Sol rated notable or better then got a Claude Opus agent that read the full TeX and searched the earlier literature, judging novelty as of the paper's own date.

First pass: three-model screening of the release

Each paper's TeX source was cut down to a digest: abstract, first ~4,000 words, every definition environment, and every sentence that emphasises a term (\emph, \textit, \textbf) or uses definitional phrasing ("we call", "is called", "we define"). Definitions in other styles beyond the first 4,000 words can be missed; those styles are more common in the arXiv papers, so this cannot explain the gap. Three models of rising cost then narrowed 719 papers to 8.

Luna is lenient and often labels named theorems as concepts. Of the 160 flagged papers, Sol rated 42 none and 114 minor.

First pass: the eight papers read in full

Opus, reading each paper in full, judged six of the eight minor; Sol's four "notable" picks survived as notable only once.

Family numbers follow the release's own catalogue (CONTENTS.md).

Sources

• openai/math repository: TeX sources, CONTENTS.md catalogue (719 manuscripts, 372 families)
• OpenAI announcement
• Pálvölgyi, A cyclic non-Ramsey heptagon, arXiv:2609.23327